Vol. 27, No. 3, 1968

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Sequences of contractions and fixed points

Sam Bernard Nadler, Jr.

Vol. 27 (1968), No. 3, 579–585

Given a convergent sequence of contraction mappings, the convergence of the sequence of their fixed points is investigated in §1 of this paper. The results obtained lead to a necessary and sufficient condition in order that a separable or a reflexive Banach space be finite dimensional. An application to differential equations is also included.

In §2 we consider mappings defined on the cartesian product of two metric spaces which are contraction mappings in one variable or in each variable separately. Using some of the results of §1 we prove that, with certain restrictions, such mappings have fixed point.

Mathematical Subject Classification
Primary: 54.85
Secondary: 34.00
Received: 9 February 1968
Revised: 22 March 1968
Published: 1 December 1968
Sam Bernard Nadler, Jr.